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arXiv:quant-ph/0606168 · Journal of Mathematical Physics

Dual monogamy inequality for entanglement

Gilad Gour, Somshubhro Bandyopadhyay, Barry C. Sanders

Published Jan 1, 2007• 133 citations
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Abstract

We establish duality for monogamy of entanglement: whereas monogamy of entanglement inequalities provide an upper bound for bipartite sharability of entanglement in a multipartite system, as quantified by linear entropy, we prove that the same quantity (namely, linear entropy) provides a lower bound for distribution of bipartite entanglement in a multipartite system. Our theorem for monogamy of entanglement is used to establish relations between bipartite entanglement that separate one qubit from the rest versus separating two qubits from the rest.

DOI
10.1063/1.2435088

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We establish duality for monogamy of entanglement: whereas monogamy of entanglement inequalities provide an upper bound for bipartite sharability of entanglement in a multipartite system, as quantified by linear entropy, we prove that the same quantity (namely, linear entropy) provides a lower bound for distribution of bipartite entanglement in a multipartite system. Our theorem for monogamy of entanglement is used to establish relations between bipartite entanglement that separate one qubit from the rest versus separating two qubits from the rest.
Abstract The distribution of entanglement in a multiparty system can be described by the principle of monogamy. Monogamy is a fundamental characteristic of entanglement that restricts its distribution among more than two parties. In this work, our aim is to explore how quantum entanglement is distributed in accordance with monogamy relations, which utilize both genuine multipartite entanglement measures and bipartite entanglement measures. Specifically, we treat the source entanglement or the accessible entanglement as genuine multipartite entanglement measures and use the entanglement of formation for bipartite cases. For GHZ class states, we analytically demonstrate that the square of the source entanglement serves as an upper bound for the sum of the squares of the entanglement of formation of the reduced subsystems, with some exceptions in specific non-generic GHZ states. We will also present numerical evidence supporting this result for W-class states. In addition, we also explore the monogamy relationship by using accessible entanglement as an upper bound. Our analysis reveals an opposite but consistent pattern in the behavior of monogamy depending on whether source entanglement or accessible entanglement is used as the upper bound. This pattern emphasizes the dual nature of these two measures.
The monogamy of entanglement is generally discussed using a bipartite entanglement measure as an upper bound. Here we discuss a new kind of monogamous relation where the upper bound is given by a multipartite measure of entanglement, the generalized concurrence. We show a monogamous equality involving the multipartite concurrence, all the bipartite concurrences, and the genuine tripartite entanglement for pure three-qubit systems. The result extends to mixed states in an inequality involving the generalized concurrence and all the bipartite concurrences. We provide a counterexample showing that the result cannot be extended for systems with more than three qubits.
In this paper, we prove a stronger entanglement monogamy inequality in a 2⊗2⊗3 system's pure state |Ψ⟩ABC. Specifically, we show that the linear entropy of ρA, which is the entanglement between A and BC, is always larger than the sum of the square of concurrence between A and B and the square of concurrence of assistance between A and C. Our proof is based on direct generalizations of the qubit system's results. Our inequality is stronger than the known monogamy inequality of concurrence and shows that the entanglement of assistance always comes from the existing entanglement. However, our inequality also shows that unlike the three-qubit case, in higher dimensional systems the entanglement between A and BC cannot be completely transformed into bipartite entanglement with assistance. Through our proof, we also give some cases when the inequality reduces to an equality.
The entanglement in multipartite quantum system is hard to characterize and quantify, although it has been intensively studied in bipartite systems. The monogamy of entanglement, as a special property of multipartite systems, shows the distribution of entanglement in the system. In this paper, we investigate the monogamy relations for multi-qubit systems. By using two entangled measures, namely the concurrence C and the negativity Nc, we establish tighter monogamy inequalities for their α-th power than those in all the existing ones. We also illustrate the tightness of our results for some classes of quantum states.
In this paper, we study the monogamy inequality of Tsallis qqq-entropy entanglement. We first provide an analytic formula of Tsallis qqq-entropy entanglement in two-qubit systems for $\frac{5\ensuremath{-}\sqrt{13}}{2}\ensuremath{\le}q\ensuremath{\le}\frac{5+\sqrt{13}}{2}.$ The analytic formula of Tsallis qqq-entropy entanglement in $2\ensuremath{\bigotimes}d$ system is also obtained and we show that Tsallis qqq-entropy entanglement satisfies a set of hierarchical monogamy equalities. Furthermore, we prove the squared Tsallis qqq-entropy entanglement follows a general inequality in the qubit systems. Based on the monogamy relations, a set of multipartite entanglement indicators is constructed, which can detect all genuine multiqubit entangled states even in the case of NNN-tangle vanishes. Moreover, we study some examples in multipartite higher-dimensional system for the monogamy inequalities.